It's in PM, not PP, that we read of Dick joining the train at Crewe. That's a year and a half after WH, so he could easily be old enough to be at public school in PM despite the evidence on his young age in WH.

However I think we have to face up to the fact that the evidence on characters' ages and sizes is not entrely consistent. For example in PM Dot has the same size "hoofs" as Peggy, whereas in S&A Peggy is the same size as John.

In PM, Dick has done qualitative analysis in chemistry at school, which I imagine would not be in a School Certificate course (the then equivalent of the later O levels/GCSE), so I guess that suggests an age of 16+.